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September 12, 2025Open Access

On the Zeros of Linear Factors of Cyclotomic Polynomials Over Galois Fields

Puntos clave

The study finds that for each pair of prime q and integer n, the sum of the roots of linear factors is always congruent to 0 modulo q.
Results reveal that the product of these roots is congruent to -1 modulo q, highlighting important modular relationships.
This analysis focuses on cyclotomic polynomial yn −1 decomposing into linear factors, particularly for prime fields with q ≤ 37.
The predictable relationships among zeros can facilitate the efficient creation of generator polynomials with specific properties.

Resumen

The study of cyclic codes via cyclotomic polynomial over Galois field has been active area of research due to their direct application in generating cyclic codes and error-correcting codes. Let q be a prime number and Fq be a given finite field with q elements. This research investigates the cyclotomic polynomial yn −1 specifically focusing on cases where yn −1 completely decomposes into linear factors over Fq for q ≤ 37 and n ≥ 2. The relationships between the field, the sum, and the product of the zeros of these linear factors are explored. The results shows that for each tested pair (q,n) where n | (q−1), the sum of the roots is always ≡ 0 (mod q), the product of the roots is ≡ −1 (mod q) and the inverse of the ratio of the product of the roots to n is ≡ q−n (mod q). The predictable modular relationships among the zeros, can be applied to the efficient design of generator polynomials with desired properties.

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